# Transformations and Distribution Function

#### JohnK

##### New Member
Hey guys,

Consider the following short example of transformations.

Let the joint density of $$x\ \text{and}\ y$$ be given by the unit square, i.e.

$$\displaystyle f_{X,Y}\left( x,y \right)=\begin{cases} 1\ 0<x<1\ \text{and}\ 0<y<1 \\ 0\ \text{elsewhere}\end{cases}$$

Then the Cumulative Distribution Function of $$z=x+y$$ is given by:

$$\displaystyle F_{Z}\left( z \right) = \begin{cases}0\ \text{for}\ z<0 \\ \int_0^z \int_0^{z-x} dydx\ \text{for}\ 0\leq z < 1 \\ 1-\int_{z-1}^1 \int_{z-x}^1 dydx\ \text{for}\ 1\leq z<2 \\ 1\ \text{for}\ 2\leq z \end{cases}\$$

I understand of course why the CDF is 0 and 1 but for the 2 cases in the middle I have been struggling to understand why we partition the CDF like that. In general what is the intuition behind the above result? I am very confused so any help is greatly appreciated, thanks.

#### BGM

##### TS Contributor
Note that the CDF is $$F_Z(z) = \Pr\{X + Y \leq z\}$$

As what you said, the support of $$(X, Y)$$ is a unit square in the $$x-y$$ plane.

And the region $$\{(x, y): x + y \leq z\}$$ is the bottom-left region divided by the line $$L: x + y = z$$ which has a slope of $$-1$$.

So you just need to know the position of $$L$$ for different values of $$z$$. It will be easier if you can draw it out (at least in your mind)

E.g. when $$z < 0$$ or $$z > 2$$, $$L$$ will not intersect with the unit square and thus you have the trivial solution.

And when $$z = 1$$, the line is just the diagonal line. Then you should understand why you need to split case at this value.

For the upper and lower limit it is just a result from "combining" the inequality, and you should be figure it out when you have the diagram.

#### JohnK

##### New Member
Thanks but I have to say that the second case, that is $$1-\int_{z-1}^1 \int_{z-x}^1 dydx$$, is still far from evident to me. Could you please be a little more specific on that one?

Also, isn't that partition arbitrary? The function is continuous so what if we chose $$z=0.5$$?

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